Limit Elastic Analysis of E-FGM Rotating Disk with Temperature Dependent Mechanical Properties

Since functional grading of materials opens new horizons in manufacturing and applications of high strength component at low weight, material tailoring plays a significant role in achieving desired material properties. In applications of high speed rotating disks, such as engine rotors, brake disks, impellers, flywheel etc., stresses in a disk can be optimized by either changing disks profile or by material grading i.e. by changing material composition. Depending upon the grading index (n) variation in FGM, it is named as L-FGM (linear functionally graded material) or P-FGM (power law functionally graded material). When material grading is exponential it is known as E-FGM and S-FGM is the abbreviation coined for sigmoid law based material grading. Kordkheili and Naghdabadi [1] examined stresses and displacement in FG disks under the influence of centrifugal force and uniform thermal load was investigated by changing volume fraction as power law for different grading indices (n) and disk responses for thermal loading on stresses were reported. For similar grading indices Bayat et al. [2] studied varying disk profiles in which concave disk profile is found to be the most light-weight disk followed by linear, convex, and uniform thickness disk. It is reported that convergent disks have lesser induced stresses compared to uniform thickness disk followed by divergent disk profiles. Nie and Batra [3] performed material tailoring to keep hoop stress or the sum of radial and hoop stress as constant is reported for arbitrary variation of the shear modulus with radius of the disk. Nejad et al. [4] performed elasto-plastic analysis of FGM disk based on power law variation of Young’s modulus, density and yield limit is reported, wherein the stresses and deformations of three different plastic regions were presented using Tresca’s yield condition. Çallıoğlu et al. [5] described similar variation in mechanical properties for different angular velocity based on von Mises yield condition. It is observed that yielding region expands throughout the outer surface of the disk for non-work hardening case under increasing angular speed.

Bhowmick et al. [6] performed limit elastic analysis of externally loaded rotating solid disk i.e. disks carrying attached masses at different locations using variational formulation method for different disk profiles and loading parameters. In another work Bhowmick et al. [7], the yield front propagation of exponential and parabolic profile annular disk and later for solid disk in Bhowmick et al. [8] under elasto-plastic regime using von Mises yield criterion for linear strain hardening material behaviour is reported using variational formulation based on Galerkin’s error minimization principle. Nayak and Saha [9] defines similar variational formulation method to study stress distribution and limit analysis for both solid and annular disk under thermo-mechanical loading for different disk geometries and temperature distributions. Zheng et al. [10], assumed similar property variation like in Kordkheili and Naghdabadi [1] and substituted volume fraction power law variation in the rule of mixture and identified the effect of the angular acceleration on FG rotating disks of same mass with non-uniform thickness on the stress distributions using von Mises stress theory.

Nakamura et al. [11] defines an empirical relation for finding Young’s modulus and yield stress variation  but the parameter associated stress-strain transfer ratio (q) was not known which restrict its application, later in Bhattacharyya et al. [12] performed experiments for different Al/SiC compositions to calculate Young’s modulus and identified stress to strain transfer ratio with Nakamura et al. [11] formulation. Cho and Ha [13] investigated parameter (q) numerically using three different averaging methods, the linear rule of mixture, MROM and the Wakashima-Tsukamoto estimate, and compared results with the finite-element discretization approach and reported (q) value as 500 GPa for Ni-Al₂O₃, later Cho and Ha et al. [14] considered same value of q and performed volume fraction optimization to minimize thermal stresses using penalty-function and golden section method.

Javaheri, R and Eslami [15] investigated thermal buckling analysis on functionally graded rectangular plate, for one dimensional non-linear temperature variation by solving a steady-state heat transfer equation. Bayat et al. [16] reported high temperature stresses by taking temperature dependent and temperature independent Young’s modulus and thermal expansion and uses rule of mixture for different disk profiles and found concave thickness disk performs better than other profiles. Sondhi et al [17] performed limit elastic analysis of FG disk for different aspect ratio taking exponential variation of Young’s modulus and density for different grading index where an existence of critical grading parameter is shown. Madan et al. [18] investigated limit elastic analysis of L-FGM using modified rule of mixture and later in Madan et al. [19] performed similar analysis for temperature dependent mechanical properties. Madan et al. [20] studied the effect of sigmoid graded disk on limit analysis and yielding locations. Zohra et al. [21] investigated power law grading effects on natural frequency on FG beams and Belhadj [22] performed free vibration for MWCNT using differential quadrature method (DQM).

In present work, to improve disk performance, limit elastic analysis was performed for temperature dependent (TD) and temperature independent (TID) mechanical properties for E-FGM rotating disks of different aspect ratio and grading indices (n). Firstly, temperature variation with radial coordinate is formulated and based on these variations, Young’s modulus of metal and ceramic were calculated. Equivalent Young’s modulus for FGM was calculated using modified rule of mixture. Effective yield stress variation for FG disk is then calculated based on temperature dependent and temperature independent Young’s modulus. The von Mises stress thus obtained is compared with yield stress to calculate the limit elastic speed of the disk. Yield stress variation is plotted along with other stresses to identify location of yielding, the knowledge of which helps in further tailoring or reinforcement of FG disks.

A uniform thickness disk of thickness (h₀) of inner and outer radius (a) and (b) is considered as shown in Figure 3. Disk rotates with an angular speed of ω. Assuming plane stress condition, centrifugal loading produces radial and tangential stresses and the resulting strain energy and external work done is given by Eq. (11) and Eq. (12).

$U=\frac{1}{2}\int\limits_{V}{\left( \sigma \varepsilon  \right)dv}=\frac{1}{2}\int\limits_{V}{\mathop{\left( {{\sigma }_{\theta }}{{\varepsilon }_{\theta }}+{{\sigma }_{r}}{{\varepsilon }_{r}} \right)dv}^{}}$ (11)

$W=-\int{u{{\omega }^{2}}rdm}$ (12)

For small displacements the total potential of the system remains unchanged, hence,

$\partial (U+W)=0$ (13)

Using constitutive and strain displacement relations, from Eq. (11), Eq. (12) and Eq. (13), we get;

$\delta \left[ \begin{align}  & \frac{\pi }{1-{{\mu }^{2}}}\int_{a}^{b}{\left\{ E(r)\frac{{{u}^{2}}}{r}+E(r)2\mu u\frac{du}{dr}+E(r)r{{\left( \frac{du}{dr} \right)}^{2}} \right\}}hdr \\ & \text{                                                   }-2\pi {{\omega }^{2}}\int\limits_{a}^{b}{\rho (r){{r}^{2}}}uhdr \\ \end{align} \right]=0$ (14)

Substituting normalized coordinate as ξ=(r-a)/(b-a), in Eq. (14) and taking $\bar{r}$= (b-a), the following is obtained.

$\delta \left[ \begin{align}  & \frac{\pi }{1-{{\mu }^{2}}}\int_{a}^{b}{E(r)\left\{ \frac{{{u}^{2}}}{\bar{r}\xi +a}+\frac{2\mu u}{{\bar{r}}}\frac{du}{d\xi }+\frac{\bar{r}\xi +a}{{{{\bar{r}}}^{2}}}{{\left( \frac{du}{d\xi } \right)}^{2}} \right\}}hd\xi  \\  & \text{                                           }-2\pi {{\omega }^{2}}\bar{r}\int\limits_{a}^{b}{\rho (r){{\left( \bar{r}\xi +a \right)}^{2}}}uhd\xi  \\ \end{align} \right]=0$ (15)

Linear polynomial function is assumed for displacement (u);

$u(\xi )=\sum{{{c}_{i}}}{{\varphi }_{i}},  i = 1,2,3,.........,{{n}_{f}}$ (16)

Gram-Schmidt scheme is employed to identify higher order orthogonal functions φ_i are the set of orthogonal polynomials. The start functions obtained satisfies the traction boundary condition of radial stress, $\left.\sigma_{r}\right|_{a}$=0 and $\left.\sigma_{r}\right|_{b}$=0. The start function thus obtained can be written as;

${{\varphi }_{o}}(r)=\frac{{{\omega }^{2}}r(3+\mu )}{8}\left[ \frac{\rho (r)}{E(r)}\left\{ \left( {{b}^{2}}+{{a}^{2}} \right)\left( 1-\mu  \right)-\frac{1+3\mu }{3+\mu }{{r}^{2}}+\frac{{{a}^{2}}{{b}^{2}}}{{{r}^{2}}}\left( 1+\mu  \right) \right\} \right]$ (17)

Putting Eq. (17) in Eq. (16) the algebraic integral form thus becomes:

$\delta \left[ \frac{\pi }{1-{{\mu }^{2}}}\int_{_{0}}^{^{1}}{\left\{ E(r)\left( \begin{align} & {{\frac{\left( \sum{{{c}_{i}}}\varphi  \right)}{\bar{r}\xi +a}}^{2}}+\frac{2\mu }{{\bar{r}}}\left[ \sum{{{c}_{i}}}\varphi \frac{d\left( \sum{{{c}_{i}}}{{\varphi }_{i}} \right)}{d\xi } \right] \\ & +\frac{\bar{r}\xi +a}{{{r}^{2}}}{{\left( \frac{d\left( \sum{{{c}_{i}}}{{\varphi }_{i}} \right)}{d\xi } \right)}^{2}} \\ \end{align} \right) \right\}}hd\xi  -2\pi {{\omega }^{2}}\int\limits_{0}^{1}{\rho (r)\left\{ {{\left( \bar{r}\xi +a \right)}^{2}}\sum{{{c}_{i}}}\varphi  \right\}}hd\xi  \right]=0$ (18)

In Eq. (18), the variational operator (δ) is replaced by $\frac{\delta}{\delta c_{j}}, j=1,2,3,4 \ldots . n_{f}$. Using Galerkin’s error minimization principle; the algebraic equation obtained is as follows:

$\frac{{\bar{r}}}{1-{{\mu }^{2}}}\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{{{c}_{i}}}}\int\limits_{0}^{1}{E(r)}\left\{ \frac{{{\varphi }_{i}}{{\varphi }_{j}}}{\bar{r}\xi +a}+\left( \frac{\mu }{{\bar{r}}}\left( \begin{align} & \varphi _{i}^{'}{{\varphi }_{j}} \\ & +{{\varphi }_{i}}\varphi _{j}^{'} \\ \end{align} \right)+\left( \frac{\bar{r}\xi +a}{{{{\bar{r}}}^{2}}} \right)\left( \varphi _{i}^{'}\varphi _{j}^{'} \right) \right) \right\}hd\xi ={{\omega }^{2}}{{b}^{3}}\sum\limits_{i=1}^{n}{\int\limits_{0}^{1}{\rho (r)\left\{ {{\left( \bar{r}\xi +a \right)}^{2}}{{\varphi }_{i}} \right\}}}hd\xi $ (19)

Eq. (19) can be expressed in matrix form and the solution of unknown coefficients is obtained numerically from {c} = [K]^-1{R} using standard IMSL subroutines and an in-house FORTRAN code. A detailed derivation and solution algorithm of same is given in Madan et al. [20].

3.png

Figure 3. Uniform thickness FG rotating disk